We compute the information metric close to \(t=0\) , and for small \(\delta \) .
Using Eq. (REF ) in Eq. (REF ), we find here
after an elaborate but straightforward computation that there is a \(\delta =0\)  component of
the QIM which reduces to the QIM of the ground state of the transverse XY model reported
in [1]}, [2]} (of which only the \(g_{hh}\)  component is relevant here, from the arguments
above). This is more appropriately related to \(g^g_{ij}\)  as discussed after Eq. (REF ).
The components proportional to powers of \(\delta \)  are the quantities of interest here and we
present them below (we drop the superscript \(e\)  in the expressions below to avoid cluttering of notation).
We get, up to the lowest orders in \(t\)  and \(\delta \) ,
\(g_{tt}\big |_{|h|<1} &=& \frac{|\gamma |\delta ^2}{2\left(1+|\gamma |\right)}~,\nonumber \\g_{tt}\big |_{|h|>1} &=& \frac{\gamma ^2 \delta ^2\left(|h|-{\mathcal {A}}\right)}{2{\mathcal {A}}\left(1-\gamma ^2\right)}~,\nonumber \\g_{hh}\big |_{|h|<1}&=& -\frac{h \gamma ^2 \delta t^4}{3 (1+|\gamma |)^2}~,\nonumber \\g_{hh}\big |_{|h|>1}&=& \pm \frac{\gamma ^2 \delta t^4 \left(\gamma ^2-|h|\left(\gamma ^2+1\right)\left({\mathcal {A}}-|h|\right)-1\right)}{3{\mathcal {A}}\left(1-\gamma ^2\right)^2}~,~~\nonumber \\g_{th}&=&\frac{t}{\delta }g_{tt}+\frac{\delta }{t}g_{hh}-\frac{1}{6}\gamma ^2 \delta t^3~,\) 
